Circle packing in a circle

Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle.

Minimum solutions:^[1]

Number of unit circles	Enclosing circle radius	Density	Optimality	Diagram
1	1	1.0000	Trivially optimal.
2	2	0.5000	Trivially optimal.
3	$1+\frac{2}{3} \sqrt{3}$ ≈ 2.154...	0.6466...	Trivially optimal.
4	$1+\sqrt{2}$ ≈ 2.414...	0.6864...	Trivially optimal.
5	$1+\sqrt{2(1+\frac{1}{\sqrt{5}})}$ ≈ 2.701...	0.6854...	Proved optimal by Graham in 1968.[2]
6	3	0.6667...	Proved optimal by Graham in 1968.[2]
7	3	0.7778...	Proved optimal by Graham in 1968.[2]
8	$1+\frac{1}{\sin(\frac{\pi}{7})}$ ≈ 3.304...	0.7328...	Proved optimal by Pirl in 1969.[3]
9	$1+\sqrt{2(2+\sqrt{2})}$ ≈ 3.613...	0.6895...	Proved optimal by Pirl in 1969.[3]
10	3.813...	0.6878...	Proved optimal by Pirl in 1969.[3]
11	$1+\frac{1}{\sin(\frac{\pi}{9})}$ ≈ 3.923...	0.7148...	Proved optimal by Melissen in 1994.[4]
12	4.029...	0.7392...	Proved optimal by Fodor in 2000.[5]
13	$2 + \sqrt{5}$ ≈4.236...	0.7245...	Proved optimal by Fodor in 2003.[6]
14	4.328...	0.7474...	Conjectured optimal.[7]
15	4.521...	0.7339...	Conjectured optimal.[7]
16	4.615...	0.7512...	Conjectured optimal.[7]
17	4.792...	0.7403...	Conjectured optimal.[7]
18	$1+\sqrt{2}+\sqrt{6}$ ≈ 4.863...	0.7611...	Conjectured optimal.[7]
19	$1+\sqrt{2}+\sqrt{6}$ ≈ 4.863...	0.8034...	Proved optimal by Fodor in 1999.[8]
20	5.122...	0.7623...	Conjectured optimal.[7]

References

1. ^ Erich Friedman, Circles in Circles on Erich's Packing Center
2. ^ ^{a b c} R.L. Graham, Sets of points with given minimum separation (Solution to Problem El921), Amer. Math. Monthly 75 (1968) 192-193.
3. ^ ^{a b c} U. Pirl, Der Mindestabstand von n in der Einheitskreisscheibe gelegenen Punkten, Math. Nachr. 40 (1969) 111-124.
4. ^ H. Melissen, Densest packing of eleven congruent circles in a circle, Geom. Dedicata 50 (1994) 15-25.
5. ^ F. Fodor, The Densest Packing of 12 Congruent Circles in a Circle, Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry 41 (2000) ?, 401–409.
6. ^ F. Fodor, The Densest Packing of 13 Congruent Circles in a Circle, Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry 44 (2003) 2, 431–440.
7. ^ ^{a b c d e f} Graham RL, Lubachevsky BD, Nurmela KJ,Ostergard PRJ. Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.
8. ^ F. Fodor, The Densest Packing of 19 Congruent Circles in a Circle, Geom. Dedicata 74 (1999), 139–145.

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